If (x + k) is a factor of f(x), which of the following must be true?

A) x = –k and x = k are roots of f(x)

B) Neither x = –k nor x = k is a root of f(x).

C) f(–k) = 0

D) f(k) = 0

On dividing two numbers, if we are getting remainder equal to zero then the smaller number will be the factor of the larger number.

Answer: Option C f(–k) = 0  is correct.

Let us proceed step by step to determine the correct option.

Explanation:

When we divide any polynomial by (x - a), we obtain a result of the form:

f (x) = (x - a) q (x) + f (a) [ From Euclid's division algorithm ] 

Here, q (x) is a polynomial with one degree less than the degree of f(x) and f(a) is the remainder. This is called the remainder theorem.

If the remainder f (a) = 0, then (x − a) is a factor of f (x)    --------(1)

Hence, from the given data in the question,

(x + k) is a factor of f(x)

Therefore, f (-k) = 0 on substituting x = -k    [ from equation (1) ]

Thus, option C f(–k) = 0  is correct.